American Institute of Mathematical Sciences

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April  2005, 1(2): 171-180. doi: 10.3934/jimo.2005.1.171

trust region method for nonsmooth convex optimization

Nobuko Sagara 1, and  Masao Fukushima 2,

1. 

Managerial Research Institute, Aichi University, Miyoshi, Aichi 470-0296, Japan
2. 

Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan

Received  April 2004 Revised  October 2004 Published  April 2005

We propose an iterative method that solves a nonsmooth convex optimization problem by converting the original objective function to a once continuously differentiable function by way of Moreau-Yosida regularization. The proposed method makes use of approximate function and gradient values of the Moreau-Yosida regularization instead of the corresponding exact values. Under this setting, Fukushima and Qi (1996) and Rauf and Fukushima (2000) proposed a proximal Newton method and a proximal BFGS method, respectively, for nonsmooth convex optimization. While these methods employ a line search strategy to achieve global convergence, the method proposed in this paper uses a trust region strategy. We establish global and superlinear convergence of the method under appropriate assumptions.
Keywords: strong convexity, inexact function and gradient evaluations., Nonsmooth convex optimization, BFGS method, Moreau-Yosida regularization, trust region method.
Mathematics Subject Classification: 90C25, 90C3.
Citation: Nobuko Sagara, Masao Fukushima. trust region method for nonsmooth convex optimization. Journal of Industrial and Management Optimization, 2005, 1 (2) : 171-180. doi: 10.3934/jimo.2005.1.171
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